Descriptive Geometry/Mathematical Constructions/Golden Ratio

Also called the golden rectangle or golden mean, the Golden Ratio has an important place in artistic, architectural, and mathematical history. It is renowned for its aesthetically pleasing proportions and the Golden Ratio consists of a square (a x a) and a smaller golden rectangle (a x b). Mathematically, the golden ratio is represented by the value φ (1.6180339887...).

${\begin{array}{l}\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\;{\overset {\text{def}}{=}}\;\phi \\\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots \end{array}}$

Method 1
Given: Long Side (AB)

1. Take the midpoint of AB (labeled M), and draw a line perpendicular along B to end at the point labeled D. Draw an arc from B connecting points M and D.

2. Draw construction line AD and draw an arc from D connect B and E (point along AD where DB = DE)

3. Draw a line from A perpendicular to AB and draw an arc from A connecting E and that line at point C.

4. Make rectangle with C, A, B, F

Method 2
Given: Short Side (AC)

1. Draw a line perpendicular to AC from point A. Draw an arc from A connecting B to the new line. Where the arc meets the line, label point E

2. Deaw a line perpendicular to AE from point E. From point E, draw an arc from point A to meet the newest perpendicular line and label the intersection with point D

3. Label the midpoint of AE with point M and draw a construction line between points M and D. Draw an arc from point M to connect the line perpendicular to AC. Label that point with B

4. Form Golden Ration from points C, A, B, F

Two Practice Problems: Question 1: http://commons.wikimedia.org/wiki/File:DescriptiveGeometry_MathematicalConstructions_GoldenRatio_Problem1.jpg Solution 1: http://commons.wikimedia.org/wiki/File:DescriptiveGeometry_MathematicalConstructions_GoldenRatio_Solution1.jpg Question 2: http://commons.wikimedia.org/wiki/File:DescriptiveGeometry_MathematicalConstructions_GoldenRatio_Problem2.jpg Solution 2: http://commons.wikimedia.org/wiki/File:DescriptiveGeometry_MathematicalConstructions_GoldenRatio_Solution2.jpg

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  Practice Problem 1
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  Solution Problem 1
• Practice Problem 2
  Practice Problem 2
• Solution Problem 2
  Solution Problem 2

Sources Cited:

Image: http://www.sandrashaw.com/AH1L18.htm

Facts: http://www.mathsisfun.com/numbers/golden-ratio.html

Information: Descriptive Geometry- Course at Carnegie Mellon University - Prof. Ramesh Krishnamurti - Provided by Carter Nelson